Also known as \(q\)-design, Geometric design.

## Description

A \(q\)-ary code that can be mapped into a subspace \(t\)-\((n,w,\lambda)_q\) design.

Subspace designs exist for all parameters in sufficiently large dimension that also satisfies divisibility constraints [3,4].

## Notes

See [5] for a review on subspace designs.Popular summary of the existence of subspace designs in Quanta Magazine.

## Parents

- Galois-field \(q\)-ary code
- Design — Subspace designs are designs on a space of fixed-weight \(q\)-ary strings (a.k.a. \(q\)-Johnson association scheme) [6].

## Child

- Combinatorial design — Combinatorial designs are designs on a space of fixed-weight binary strings (a.k.a. Johnson association scheme) [7,8]. Subspace designs reduce to combinatorial designs for \(q=2\).

## References

- [1]
- Cameron, Peter J. "Generalisation of Fisher’s inequality to fields with more than one element." Combinatorics, London Math. Soc. Lecture Note Ser 13 (1973): 9-13.
- [2]
- V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing (2013) DOI
- [3]
- A. Fazeli, S. Lovett, and A. Vardy, “Nontrivial t-designs over finite fields exist for all t”, Journal of Combinatorial Theory, Series A 127, 149 (2014) DOI
- [4]
- P. Keevash, A. Sah, and M. Sawhney, “The existence of subspace designs”, (2023) arXiv:2212.00870
- [5]
- M. Braun, M. Kiermaier, and A. Wassermann, “q-Analogs of Designs: Subspace Designs”, Network Coding and Subspace Designs 171 (2018) DOI
- [6]
- Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI
- [7]
- Delsarte, Philippe. "An algebraic approach to the association schemes of coding theory." Philips Res. Rep. Suppl. 10 (1973): vi+-97.
- [8]
- V. I. Levenshtein, “Universal bounds for codes and designs,” in Handbook of Coding Theory 1, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp.499-648.

## Page edit log

- Victor V. Albert (2024-01-09) — most recent

## Cite as:

“Subspace design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subspace_design